It is standard to view the graded PH module as a quiver representation of the directed graph \(0\rightarrow \cdots \rightarrow N\) . A celebrated theorem of Gabriel [1]} states that every such representation decomposes as a direct sum of indecomposable pure-gradedBy definition, an element \(v\)  of a graded vector space \(V = \bigoplus _p V^p\)  is pure-graded if \(v \in V^p\)  for some \(p\) . sub-representations. Concretely, this means that for each homological grading \(n\) 

Pn( F C )
=
V V V


for some family of submodules \({V}\) , where each \(V = (V^0\rightarrow \cdots \rightarrow V^N) \in {V}\)  is isomorphic to an interval representation of the form
\(0\rightarrow \cdots \rightarrow 0 \rightarrow \mathbb {K}\xrightarrow{} \cdots \xrightarrow{} \mathbb {K}\rightarrow 0 \rightarrow \cdots \rightarrow 0.\) 
